\newproblem{lay:1_2_34}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 1.2.34}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
  In a wind tunnel, the force on a projectile due to air resistance was measured at different velocities:\\
	\begin{center}
		\begin{tabular}{lcccccc}
			Velocity (100 ft/s) & 0 & 2 & 4 & 6 & 8 & 10 \\ 
			Force (100 lb)      & 0 & 2.90 & 14.8 & 39.6 & 74.3 & 119 \\
		\end{tabular}
	\end{center}
	Find an interpolating polynomial for these data and estimate the force on the projectile when it is travelling at 750 ft/s.
	Use $f(t)=a_0+a_1t+a_2t^2+a_3t^3+a_4t^4+a_5t^5$. What happens if you try to use a polynomial of degree 3?
}
{
  % Solution
	Similarly to the previous problem, the equation system is 
	\begin{center}
		$\left(\begin{array}{cccccc}1 & 0 & 0^2 & 0^3 & 0^4 & 0^5 \\ 1 & 2 & 2^2 & 2^3 & 2^4 & 2^5 \\ 1 & 4 & 4^2 & 4^3 & 4^4 & 4^5 \\
		   1 & 6 & 6^2 & 6^3 & 6^4 & 6^5 \\ 1 & 8 & 8^2 & 8^3 & 8^4 & 8^5 \\ 1 & 10 & 10^2 & 10^3 & 10^4 & 10^5 \end{array}\right)
		\left(\begin{array}{c}a_0\\a_1\\a_2\\a_3\\a_4\\a_5\end{array}\right)=
		\left(\begin{array}{c}0\\2.9\\14.8\\39.6\\74.3\\119\end{array}\right)
		$
	\end{center}
	Its solution is
	\begin{center}
		$f(t)=1.7125t-1.1948t^2+0.6615t^3-0.0701t^4+0.0026t^5$
	\end{center}
	At a velocity of 750 ft/s, the force on the projectile is
	\begin{center}
		$f(7.50)=1.7125(7.50)-1.1948(7.50)^2+0.6615(7.50)^3-0.0701(7.50)^4+0.0026(7.50)^5=64.6 (100 lb)$
	\end{center}
	
	If we try to solve the same equation system with a polynomial of degree 3,
	\begin{center}
		$\left(\begin{array}{cccccc}1 & 0 & 0^2 & 0^3 \\ 1 & 2 & 2^2 & 2^3 \\ 1 & 4 & 4^2 & 4^3 \\
		   1 & 6 & 6^2 & 6^3 \\ 1 & 8 & 8^2 & 8^3\\ 1 & 10 & 10^2 & 10^3 \end{array}\right)
		\left(\begin{array}{c}a_0\\a_1\\a_2\\a_3\end{array}\right)=
		\left(\begin{array}{c}0\\2.9\\14.8\\39.6\\74.3\\119\end{array}\right)
		$
	\end{center}
	we find that there is no solution of the equation system.
}
\useproblem{lay:1_2_34}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
